2013 Dolomites Research Week on Approximation
Lecture 7:
Kernel methods for more
general surfaces
Grady B. Wright
Boise State University
Overview
•  Background
•  Kernel approximation on surfaces
•  Applications to numerically solving PDEs on surfaces
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Interpolation with kernels
Examples:
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Interpolation with kernels
Examples:
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Kernel interpolation on surfaces
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•  Kernels on the sphere:
o  Schoenberg (1942)
o  See Lecture 1 slides for more…
•  Kernels on specific manifolds (SO(3), motion group,
projective spaces):
o  Erb, Filber, Hangelbroek, Schmid, zu Castel,…
•  Kernels on arbitrary Riemannian manifolds:
o  Narcowich (1995)
o  Dyn, Hangelbroek, Levesley, Ragozin, Schaback, Ward,
Wendland.
•  In these studies the kernels used are highly dependent on the
manifold.
o  Inherent benefits to this.
o  However, for arbitrary manifolds it is difficult (or impossible) to
compute these kernel.
Kernel interpolation on surfaces
•  Types of surfaces:
•  Examples:
•  Applications:
o  geophysics
o  atmospheric sciences
o  biology
o  chemistry
o  computer graphics
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Kernel interpolation on surfaces
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Kernel interpolation on surfaces
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Nodes on surfaces
•  Kernel methods do not require a mesh, just a set of nodes.
•  Some ideas:
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Motivation: Reaction diffusion equations on surfaces
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•  Prototypical model: 2 interacting species
•  Applications
o  Biology: diffusive transport on a membrane, pattern formation on animal
coats, and tumor growth.
o  Chemistry: waves in excitable media (cardiac arrhythmia, electrical
signals in the brain).
o  Computer graphics: texture mapping and synthesis and image processing.
Current methods and kernel-based approach
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•  Current numerical method can be split into 2 categories:
1. Surface-based: approximate the PDE on the surface using intrinsic coordinates.
Triangulated Mesh
Dziuk (1988)
Stam (2003)
Xu (2004)
Dziuk & Elliot (2007)
Logically rectangular grid
Calhoun and Helzel (2009)
2. Embedded: approximate the PDE in the embedding space, restrict solution to surface.
Level Set
Bertalmio et al. (2001)
Schwartz et al. (2005)
Greer (2006)
Sbalzarini et al. (2006)
Dziuk & Elliot (2010)
Closest point:
Ruuth & Merriman (2008)
MacDonald & Ruuth (2008)
MacDonald & Ruuth (2009)
•  Kernel-based method: Fuselier & W (2013)
•  Similarity to 1: approximate the PDE on the surface.
•  Similarity to 2: use extrinsic coordinates.
•  Differences: method is mesh-free;
computations done in same dimension as the surface.
Interpolation with restricted PD kernels
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Interpolation with restricted PD kernels
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Main idea: Trace theorem and restriction and extension operators on
the native space from Schaback (1999).
Interpolation error estimates
•  Specific error estimate results from Fuselier & W (2012).
o  More general results are given in the paper.
Notation:
Theorem: target functions in the native space
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Interpolation error estimates
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•  Specific error estimate results from Fuselier & W (2012).
o  More general results are given in the paper.
Notation:
Corollary: target functions approx. twice as smooth as the native space
Interpolation error estimates
•  Specific error estimate results from Fuselier & W (2012).
o  More general results are given in the paper.
Notation:
Theorem: target functions rougher than the native space
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Interpolation error estimates
•  Specific error estimate results from Fuselier & W (2012).
o  More general results are given in the paper.
Notation:
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Return: Reaction diffusion equations on surfaces
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•  Prototypical model: 2 interacting species
•  Applications
o  Biology: diffusive transport on a membrane, pattern formation on animal
coats, and tumor growth.
o  Chemistry: waves in excitable media (cardiac arrhythmia, electrical
signals in the brain).
o  Computer graphics: texture mapping and synthesis and image processing.
Differential operators on surfaces
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Differential operators on surfaces
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Kernel approximation of surface derivative operators
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Kernel approximation of surface derivative operators
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Kernel approximation of surface derivative operators
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Example: convergence of discrete surface Laplacian
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•  Error estimates given in Fuselier & W (2013)
•  Observed convergence rate is 2 orders higher
than theory predicts.
Applications: Turing patterns
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•  Pattern formation via non-linear reaction-diffusion systems; Turing (1952)
Possible mechanism for animal coat formation (and other morphogenesis phenomena)
•  Example system: Barrio et al. (1999)
•  These types of systems have been studied extensively in planar domains.
•  Recent studies have focused on the sphere.
•  Growing interest in studying these on more general surfaces.
•  Numerical method: collocation and method-of-lines
(like method from Tutorials 4-6)
Application: Turing patterns
•  Surfaces used in the numerical experiments:
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Application: Turing patterns
•  Node sets X used in the numerical experiments:
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Application: Turing patterns
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•  Numerical solutions: steady spot patterns (visualization of u component)
Initial condition: u and v set to random values between +/- 0.5
Application: Turing patterns
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•  Numerical solutions: steady stripe patterns (visualization of u component)
Initial condition: u and v set to random values between +/- 0.5
Application: spiral waves in excitable media
•  Example system: Barkley (1991)
u=activator species
v=inhibitor species
Simplification of FitzHugh-Nagumo model for a spiking neuron.
•  Studied extensively on planar regions and somewhat on the sphere.
•  Growing interest more physically relevant domains like surfaces.
•  Snapshots from different numerical simulations with our method:
visualization of the u (activator) component DRWA 2013
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Application: spiral waves in excitable media
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Other applications
•  The discrete approximation to the surface Laplacian can also be used
approximate the surface harmonics.
•  Question: Can one hear the shape of a Bretzel?
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Current work
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Summary and future
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Thank you to the organizers
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